GENERAL I ARTICLE Nilakantha, Euler and 1t Shailesh A Shirali Introduction In this article, we present brief expository accounts of two well-known series· for 7r: the Gregory-N{laka1J.tha series, 1r 1 1 1 1 -=1--+---+-_· .. (1) 43579 Shailesh ShiraH has been also known as Leibnitz's series, and the series Ln 1/n2, at the Rishi Valley School w hose sum was found by Leonhard Euler in 1734: (Krishnamurti Foundation oflndia), Rishi Valley, Andhra Pradesh, for ~ore (2) than ten years and is currently the Principal. He The first series is usually known as the Gregory-Leibnitz se­ has been involved in the ries but recent findings have shown that it was known to Mathematical Olympiad mathematicians in Kerala well before Gregory and Leibnitz Programme since 1988. He or the invention of the calculus. Evidently the discoveries has a deep interest in were made independent of one another. talking and writing about mathematics, particularly about its historical aspects. He is also Preliminaries on Infinite Series interested in problem solving (particularly in the Infinite series are fun to play with-often very pretty iden­ fields of elementary tities emerge from their study. Sometimes they are easy to number theory, geometry evaluate. We have, for instance, 1/2+1/4+1/8+1/16+·· _ and combinatorics). 1, and more generally, x -l-x- -- for Ixl < 1. (3) 1 The well known paradox of Geometric series of this type are well known at the high­ Achilles and the tortoise is 1 school level, and we shall not say more about them. Another resolved by studying the class of series that are easy to evaluate are the ones that yield underlying geometric series. R-E-S-O-N-A-N-C-E--I-M-a-Y--19-9-7---------------~~--------------------------------------2-' GENERAL I ARTICLE to partial fraction decomposition followed by telescoping. For instance, the series 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + is summed as' shown below: 1 1 1 00 1 -+-+-+ ,. = 2 . 6 12 ]; n(n + 1) = 00 (1 1) ]; ~- n+l = (i-D+G-D+G-D+· .. = 1, (4) after a wholesale cancellation of fractions. More challenging is the evaluation of the series I:n 1/n(n+l)(n+2), A partial fraction decomposition yields: 1 111 n(n + l)(n + 2) = 2(n + 2) - n + 1 + 2n (5) and a rearrangement of terms gives 00 1 1 (6) ]; n(n + l)(n + 2) = 4' (Comment: We must mention here that rearranging the terms of an infinite convergent series in order to compute its sum is not always a valid step. For instance, the series 1-1/2 + 1/3 - 1/4+, . , converges, yielding a sum of In 2, but by rearranging the terms in different ways we get different sums, For instance, when rearranged as shown below, 11111111 1+3-2+5+7-4+9+11-6+ with two positive terms followed by one negative term, the series converges to In2J2, and the series 11111111 1 + 3 + 5 + 7 - "2 + 7 + 9 + 11 - 4 + with three positive terms followed by one negative term, con­ verges to In 2V3. The notion of absolute convergence helps -30----------------------------~--------------R-E-SO--N-AN-C-E--I-M-a-Y-1-99-7 GENERAL I ARTICLE to make sense of these facts. Suppose that the series Ei ai A theorem of converges. Then the series converges 'absolutely' if ~i lail Riemann's converges. A theorem of Riemann's (extremely surprising at (extremely first encounter) states that if ~i ai converges but not abso­ surprising at first lutely (that is, ~i lail diverges), then-by suitably rearranging the terms we can get the resulting series to converge to any encounter) states desired number whatever! If the ai are all of one sign, then that if ~iaj the notions of convergence and absolute convergence coin­ converges but not cide. In the example at hand, rearrangement can be avoided absolutely (that is, by observing that ~j I aj I diverges), 1 1( 1 1) then by suitably n(n + 1)(n + 2) = 2" n(n + 1) - (n + 1)(n + 2) rearranging the terms we can get Use of the telescoping principle now yields (6).) the resulting series Another pretty result is provided by the summation to converge to any (7) desired number whatever! which holds whenever Ix I < 1. However a very different proposition is the alter)lating series 1 1 1 1 1--+---+-- (8) 3 579 It is not hard to show that the series converges, for by com­ bining pairs of terms it can be written as (9) 2 Since 4n - 1 > n(n - 1) for n ~ 1, it follows that 1 1 1 4 ~ <-+~ -- L....J 4n 2 - 1 3 L....i n (n - 1) - 3' n~l n~2 Thus the series has a well-defined sum. What is the sum? The 'school-boy' evaluation (yery likely the method used by Gregory) proceeds thus: since 2 6 8 1 1 - x + x4 - x + x - .. = -- for Ixl < 1 (10) 1 +x2 ' R-E-S-O-N-A-N-C-E-I-M-a-Y-1-9-97---------------~-------------------------------------~ GENERAL I ARTICLE we deduce, by integration between 0 and t (It I < 1), that t 3 t 5 t 7 tan-1 t = t - - + - - - + ... (11) 357 Substituting the value t = 1 in (11), we obtain the required sum as 7r /4. However the question of convergence is left open in this summation (an infinite series cannot always be integrated term-by-term in this fashion), and we need to fill in the necessary details. This is accomplished as follows. We consider the algebraic identity (note that this is not an infinite sum) 1 2 4 --=I-x+x- 1 +x2 X 2n+2 (_I)n+l_-. (12) 1 +x2 Integrating both sides between 0 and 1, we obtain 7r 1 1 1 4" = 1 - "3 + 5 + + (-1) n 2n + 1 + 1 2n+2 ( -1) n+l _x_ dx. (13) 10o 1 + x 2 The integral on the right side tends to 0 as n ~ 00, because 1 1 --dxx2n+2 1 < 10 x 2n+2 dx = 1 , (14) 2 0 110o 1 + x 2n + 3 \Yhich tends to 0 as n ~ 00. The result follows. Another approach is via evaluation of the integral In defined by f1r/4 n In = Jo tan X dx (15) for positive integers n. (This' integral is routinely encoun­ tered while studying integration by parts). We claim that In ~ 0 as n --+ 00. For proof we argue as follows. Obvi­ ously In> O. Next, the tangent function is convex over the open interval (0, 7r /4), so the chord connecting the points (0,0) and (7r /4,1) lies entirely above the tangent curve. The equation of this line is y = (4/7r}x, so it follows that -32------------------------------~------------------RE-S-O-N-A-N-C-E-I-M--aY--19-9-7 GENERAL I ARTICLE tan x < (4/1r)x for all x in the open interval (0, 1r /4). There­ fore / / 0 < In = fo 4 tannxdx < (4f- f 4 xndx 7r 0 = (if(~r+l-1 7r 4 n+l 7r (16) = 4(n + 1)' and the result follows (let n --+ 00). Next, integration by parts yields 1 In = -- - In-2· (17) n-1 From (17) and the fact that 10 = 1r / 4, we deduce that 1 1 1--+-- (18) 3 5 Letting n --+ 00, we obtain the Gregory-NIlakaI,l~ha series. This approach yields an unexpected bonus: (since 11 = Insec1r /4 = (1/2) In2, we obtain 1 1 1 1 In2=1- 2+ 3- 4+ 5- (19) (Exercise: Prove (19).) 2 Or so we think ! The facts are unfortunately nottoo well known. In the text Yuktibhasa, the series The Gregory-NIlaka~~ha Series is credited to Madhava of Both proofs presented above make use of calculus, and it Sangamagrama (1350-14101, would seem difficult to arrive at the stated result without us­ who lived almost a full century ing integration. However the Kerala school of mathematics before Nilakar)~a The result that flourished in the fifteenth and sixteenth centuries seems is stated as follows: if c is the circumference of a circle of to have been in possession of just such a proof, due origi­ diameter d, then nally to the mathematician NIlakaI,l~ha2(1444-1550). We 411 411 411 411 c=4d--+---+-- .... now present this elegant proof. In retrospect it has a cer­ 3 5 7 9 tain naturalness to it, for 7r is after all connected to the See S Balachandra Rao in circumference of a circle. Though calculus is not used at Suggested Reading for further any stage in the proof, the intuitive notion of limit is clearly details. present in the author's mind. -R-ES-O-N-A-N-C-E-I--M-a-y-19-9-7--------------~-----------------------------~- GENERAL I ARTICLE In Figure 1 OABC is a square of side 1, and AC is a C~;:;::==-____""'IlB circular arc of unit radius. The length of the arc AC is thus Q 1r 12. Let P, Q be two points on the side AB, the points occurring in the order shown (A, P, Q, B), with P, Q very p close to one another, i.e., with PQ ~ 1. Let OP and OQ meet the arc at Land M, respectively, and let P Rand L N be the perpendiculars from P and L upon line OQ. Then ~OLN tV ~OPR and ~QRP tV ~QAO. Therefore LN PR and PR = OA. (20) Figure 1 OL = OP' PQ OQ These equations give, since 0 L = 0 A = 1 PQ LN = OP x OQ (21) Now OP x OQ ~ OQ2 = 1 + AQ2, and arc LM ~ LN.